Since \(\zeta \subset \mathcal {I}\) , we have \(\zeta (x)=\zeta (T(x))\)  and \(T^{-1}\zeta =\zeta \) . Since, as showed in the proof of Lemma REF , \(\eta \)  and \(\mathcal {P}\)  satisfy (REF ), we may deduce that
\(T^{-n}(\eta \vee \zeta (T^n(x)))=\eta \vee \zeta (x)\cap \mathcal {P}^{n}(x),\) 
and
\(\mathcal {P}^{n}(T(x))\cap \eta (T(x))\cap \zeta (T(x))=\, T(\mathcal {P}^{n+1}(x))\cap \eta (T(x))\cap \zeta (T(x)),\) 
meaning that \(\eta \vee \zeta \)  also satisfies (REF ). By Proposition REF  we have for \(\Theta \) -a.e. \(x\in X\) ,
\(\lim _{n\rightarrow \infty }\frac{-\log \Theta ^{\eta \vee \zeta }_x(\mathcal {P}^{n}(x))}{n}=h_\Theta (\mathcal {P}|\widehat{\eta \vee \zeta })(x).\) 
By the uniqueness of conditional measures (Theorem REF ), we have for \(\Theta \) -a.e. \(x\in X\) ,
\(\Theta ^{\eta \vee \zeta }_x=(\Theta ^{\eta }_x)^\zeta _x=(\Theta ^{\zeta }_x)^\eta _x.\) 
We may reinterpret (REF ) as for \(\Theta \) -a.e. \(y\in X\) , for \(\Theta ^\eta _{y}\) -a.e. \(z\in \eta (y)\) , for \((\Theta ^\eta _{y})^\zeta _{z}\) -a.e. \(x\in \eta (y)\cap \zeta (z)\) ,
\(\lim _{n\rightarrow \infty }\frac{-\log (\Theta ^{\eta }_{x})^{\zeta }_{x}(\mathcal {P}^{n}(x))}{n}=h_\Theta (\mathcal {P}|\widehat{\eta \vee \zeta })(x).\) 
Note that for \(x\in \eta (y)\cap \zeta (z)\) , \(\eta (x)=\eta (y)\)  and \(\zeta (x)=\zeta (z)\) , therefore for \(\Theta ^\eta _{y}\) -a.e. \(z\in \eta (y)\) , for \((\Theta ^\eta _{y})^\zeta _{z}\) -a.e. \(x\in \eta (y)\cap \zeta (z)\) ,
\((\Theta ^{\eta }_x)^{\zeta }_x=(\Theta ^{\eta }_{y})^{\zeta }_{z}.\) 
Thus we obtain that for \(\Theta \) -a.e. \(y\in X\) , for \(\Theta ^\eta _{y}\) -a.e. \(z\in \eta (y)\) , for \((\Theta ^\eta _{y})^\zeta _{z}\) -a.e. \(x\in \eta (y)\cap \zeta (z)\) ,
\(\lim _{n\rightarrow \infty }\frac{-\log (\Theta ^{\eta }_{y})^{\zeta }_{z}(\mathcal {P}^{n}(x))}{n}=h_\Theta (\mathcal {P}|\widehat{\eta \vee \zeta })(x).\) 
Using (REF ) we deduce that for \(\Theta \) -a.e. \(y\in X\) , for \(\Theta ^\eta _{y}\) -a.e. \(z\in \eta (y)\) , for \((\Theta ^\eta _{y})^\zeta _{z}\) -a.e. \(x\in \eta (y)\cap \zeta (z)\) ,
\(\lim _{n\rightarrow \infty }\frac{\log (\Theta ^{\eta }_{y})^{\zeta }_{z}(\mathcal {P}^{n}(x)) }{\log |\mathcal {P}^{n}(x)\cap \eta (x)|_{d_X}}=\frac{h_\Theta (\mathcal {P}|\widehat{\eta \vee \zeta })(x)}{-\log \rho }.\) 
By the fact that \((\Theta ^\eta _{y})^\zeta _{z}\)  is a conditional measure of \(\Theta ^\eta _{y}\) , hence, by the definition of lower Hausdorff dimension, their essential local dimension can not be greater than the lower Hausdorff dimension of \(\Theta ^\eta _{y}\) , which is \(\frac{\alpha h_\mu +h_\nu }{-\log \rho }\)  by Lemma REF , we deduce that for \(\Theta \) -a.e. \(x\in X\) ,
\(\alpha h_\mu +h_\nu \ge h_\mu (\mathcal {P}|\widehat{\eta \vee \zeta })(x).\) 
Next we will show that the reverse inequality also holds. Using the fact that for \(\Theta ^\zeta _x\) -a.e. \(y\in \zeta (x)\) , \(\Theta ^\zeta _{y}=\Theta ^\zeta _{x}\) , we deduce that for \(\Theta \) -a.e. \(x\in X\) ,
\(h_\Theta (\mathcal {P}|\widehat{\eta \vee \zeta })(x)&=\ \mathbb {E}_\Theta ( -\log \Theta ^{\eta \vee \zeta }_\cdot (\mathcal {P}(\cdot )) |\mathcal {I})(x)\\&=\ \int _{\zeta (x)} -\log (\Theta ^{\zeta }_{y})^{\eta }_{y}(\mathcal {P}(y)) \, \Theta ^\zeta _x(\mathrm {d}y)\\&=\ \int _{\zeta (x)} -\log (\Theta ^{\zeta }_x)^{\eta }_{y}(\mathcal {P}(y)) \, \Theta ^\zeta _x(\mathrm {d}y)\\&=\ H_{\Theta ^\zeta _x}(\mathcal {P}|\widehat{\eta }).\) 

By [1]} (collected from [2]}), and that \(T^{-1}\eta =\eta \vee \mathcal {P}\) , for \(n\ge 2\)  we have
\(H_{\Theta _x^\zeta }(\mathcal {P}^{n}|\widehat{\eta })&=\ H_{\Theta _x^\zeta }(\mathcal {P}\vee T^{-1}(\mathcal {P}^{n-1})|\widehat{\eta })\\&=\ H_{\Theta _x^\zeta }(\mathcal {P}|\widehat{\eta })+H_{\Theta _x^\zeta }(T^{-1}(\mathcal {P}^{n-1})|\widehat{\eta \vee \mathcal {P}} )\\&=\ H_{\Theta _x^\zeta }(\mathcal {P}|\widehat{\eta })+H_{\Theta _x^\zeta }(T^{-1}(\mathcal {P}^{n-1})|\widehat{T^{-1}\eta } )\\&=\ H_{\Theta _x^\zeta }(\mathcal {P}|\widehat{\eta })+H_{\Theta _x^\zeta }(T^{-1}(\mathcal {P}^{n-1})|T^{-1}\widehat{\eta } )\\&=\ H_{\Theta _x^\zeta }(\mathcal {P}|\widehat{\eta })+H_{\Theta _x^\zeta }(\mathcal {P}^{n-1}|\widehat{\eta } ),\) 

where in the last step we have used the fact that \(\Theta _x^\zeta \)  is \(T\) -invariant. By iteration this implies that for \(\Theta \) -a.e. \(x\in X\)  and \(n\ge 1\) ,
\(H_{\Theta _x^\zeta }(\mathcal {P}^{n}|\widehat{\eta })=nH_{\Theta _x^\zeta }(\mathcal {P}|\widehat{\eta }).\) 
Since \(-\log \)  is a convex function, by Jensen's inequality and Theorem REF  we have
\(H_{\Theta _x^\zeta }(\mathcal {P}^{n}|\widehat{\eta })&=\ \int _X -\log (\Theta _x^\zeta )^\eta _{y}(\mathcal {P}^{n}(y)) \,\Theta _x^\zeta (\mathrm {d}y)\\&\ge \ -\log \int _X (\Theta _x^\zeta )^\eta _{y}(\mathcal {P}^{n}(y)) \,\Theta _x^\zeta (\mathrm {d}y)\\&= \ -\log \int _X \sum _{B\in \mathcal {P}^{n}} \mathbf {1}_B(y)(\Theta _x^\zeta )^\eta _{y}(B) \,\Theta _x^\zeta (\mathrm {d}y)\\&= \ -\log \sum _{B\in \mathcal {P}^{n}} \int _X (\Theta _x^\zeta )^\eta _{y}(B) \,\Theta _x^\zeta (\mathrm {d}y)\\&= \ -\log \sum _{B\in \mathcal {P}^{n}} \Theta _x^\zeta (B)\\&= \ -\log \Theta _x^\zeta (\mathcal {P}^{n}(x)),\) 

where we have used the fact that \((\Theta _x^\zeta )^\eta _{y}(B)=0\)  if \(y\notin B\)  and \(\Theta _x^\zeta (B)=0\)  if \(x\notin B\) . Therefore for \(\Theta \) -a.e. \(x\in X\)  and \(n\ge 1\) ,
\(H_{\Theta _x^\zeta }(\mathcal {P}|\widehat{\eta })\ge \frac{ -\log \Theta _x^\zeta (\mathcal {P}^{n}(x))}{n}.\) 
In other words, for \(\Theta \) -a.e. \(x\in X\) , for \(\Theta ^\zeta _x\) -a.e. \(y\in \zeta (x)\)  and \(n\ge 1\) ,
\(H_{\Theta _{x}^\zeta }(\mathcal {P}|\widehat{\eta })=H_{\Theta _{y}^\zeta }(\mathcal {P}|\widehat{\eta })\ge \frac{ -\log \Theta _{y}^\zeta (\mathcal {P}^{n}(y))}{n}=\frac{ -\log \Theta _{x}^\zeta (\mathcal {P}^{n}(y))}{n},\) 
where we have used again the fact that for \(\Theta ^\zeta _x\) -a.e. \(y\in \zeta (x)\) , \(\Theta ^\zeta _{y}=\Theta ^\zeta _x\) . By applying the SMB theorem to the ergodic measure \(\Theta ^\zeta _x\)  we deduce that for \(\Theta \) -a.e. \(x\in X\) ,
\(H_{\Theta _x^\zeta }(\mathcal {P}|\widehat{\eta })\ge h_{\Theta _x^\zeta }(T,\mathcal {P}).\) 
On the other hand, by Jacob theorem (see [3]} for example) we have that
\(\int _X h_{\Theta _x^\zeta }(T,\mathcal {P}) \, \Theta (\mathrm {d}x)=h_\Theta (T,\mathcal {P}).\) 
Recall that \(\Theta =\ell \times \mu \times \nu \)  where \(\mu \)  (resp. \(\nu \) ) is the natural extension of \(\mu ^+\)  (resp. \(\nu ^+\) ) to \(\Sigma _a\)  (resp. \(\Sigma _b\) ), for \(x=(t,\mathbf {v})=(t,\mathbf {i},\mathbf {j})\) ,
\(\Theta (\mathcal {P}^{n}(x))&=\ \Theta (\mathcal {Q}^{n}(t)\times \Sigma _{a,b}^-\times [\mathbf {v}^+|_{n,t}])\\&=\ \ell (\mathcal {Q}^{n}(t))\cdot \mu (\Sigma _a^-\times [\mathbf {i}^+|_{\tau _n(t)}])\cdot \nu (\Sigma _b^-\times [\mathbf {j}^+|_n])\\&=\ \ell (\mathcal {Q}^{n}(t))\cdot \mu ^+([\mathbf {i}^+|_{\tau _n(t)}])\cdot \nu ^+([\mathbf {j}^+|_n]).\) 

Therefore, by using SMB Theorem for the ergodic measures \(\ell \) , \(\mu ^+\)  and \(\nu ^+\)  respectively, we have for \(\Theta \) -a.e. \(x=(t,\mathbf {i},\mathbf {j})\in X\) ,
\(\lim _{n\rightarrow \infty }\frac{-\log \Theta (\mathcal {P}^{n}(x))}{n}&=\, \lim _{n\rightarrow \infty }\frac{-\log \ell (\mathcal {Q}^{n}(t))\cdot \mu ^+([\mathbf {i}^+|_{\tau _n(t)}])\cdot \nu ^+([\mathbf {j}^+|_n]) }{n}\\&=\, \lim _{n\rightarrow \infty }\frac{-\log \ell (\mathcal {Q}^{n}(t))}{n}+\frac{\tau _n(t)}{n}\frac{-\log \mu ^+([\mathbf {i}^+|_{\tau _n(t)}])}{\tau _n(t)}+ \frac{-\log \nu ^+([\mathbf {j}^+|_n])}{n} \\&=\, h_\ell (R,\mathcal {Q})+\alpha h_{\mu }+h_{\nu }\\&=\, \alpha h_{\mu }+h_{\nu },\) 

where we have used the well-known fact that \(h_\ell (R,\mathcal {Q})=0\) , since \(R\)  is an irrational rotation and \(\mathcal {Q}\)  is a partition of \(\mathbb {T}\)  consisting of two intervals (see [4]} for example). Finally by using Theorem REF  again for \(\Theta \) , this implies that
\(h_\Theta (T,\mathcal {P})= \alpha h_{\mu }+h_{\nu }.\) 
Together with (REF ), (REF ), (REF ) and (REF ) we deduce that for \(\Theta \) -a.e. \(x\in X\) ,
\(h_\mu (\mathcal {P}|\widehat{\eta \vee \zeta })(x)=\alpha h_{\mu }+h_{\nu }.\) 
By (REF ) this yields the conclusion.
